In the present paper we consider several measures Ior the risk that is present in ai1 iiisurance ... more In the present paper we consider several measures Ior the risk that is present in ai1 iiisurance environmeiit. We looi< for clesirable properties for two types of risk ineasures, the olies reflectiiig both negative and positive results, and the ineasures for insolvency risks dealing with aspects of niin, as wel1 as their relatioii to the allocation of ecoiioiiiic capita1 to different brisiness liiies or to the differeiit subcompanies constituting a financial conglomerate. The main probleili for both types of measureii~ents is that the dependence structure that exists betweeii tlie different units involved is uilknown.
In the recent actuarial literature, several proofs have been given for the fact that if a random ... more In the recent actuarial literature, several proofs have been given for the fact that if a random vector (X 1 , X 2 , . . . , X n ) with given marginals has a comonotonic joint distribution, the sum X 1 + X 2 + · · · + X n is the largest possible in convex order. In this note we give a lucid proof of this fact, based on a geometric interpretation of the support of the comonotonic distribution.
A subject often recurring in financial papers, is the pricing of stocks and securities when the r... more A subject often recurring in financial papers, is the pricing of stocks and securities when the rate of return is stochastic. In most cases, the stocks considered are assumed not to pay out any dividend. In the present contribution we want to show how it is possible to obtain upper and lower bounds for the (distribution of the) accumulated value of a dividend paying security at a future time t, when the logarithm of the stock price is modelled by means of a Wiener process.
Comparing risks is the very essence of the actuarial profession. This chapter offers mathematical... more Comparing risks is the very essence of the actuarial profession. This chapter offers mathematical concepts and tools to do this, and derives some important results of non-life actuarial science. There are two reasons why a risk, representing a non-negative random financial loss, would be universally preferred to another. One is that the other risk is larger, see Section 7.2, the second is that it is thicker-tailed (riskier), see Section 7.3. Thicker-tailed means that the probability of extreme values is larger, making a risk with equal mean less attractive because it is more spread and therefore less predictable. We show that having thicker tails means having larger stop-loss premiums.We also show that preferring the risk with uniformly lower stop-loss premiums describes the common preferences between risks of all risk averse decision makers. From the fact that a risk is smaller or less risky than another, one may deduce that it is also preferable in the mean-variance order that is ...
Analysis and Applications in the Social Sciences, 2004
ABSTRACT Multiple linear regression is the most widely used statistical technique in practical ec... more ABSTRACT Multiple linear regression is the most widely used statistical technique in practical econometrics. In actuarial statistics, situations occur that do not fit comfortably in that setting. Regression assumes normally distributed disturbances with a constant variance around a mean that is linear in the collateral data. In many actuarial applications, a symmetric normally distributed random variable with a variance that is the same whatever the mean does not adequately describe the situation. For counts, a Poisson distribution is generally a good model, if the assumptions of a Poisson process described in Chapter 4 are valid. For these random variables, the mean and variance are the same, but the datasets encountered in practice generally exhibit a variance greater than the mean. A distribution to describe the claim size should have a thick right-hand tail. The distribution of claims expressed as a multiple of their mean would always be much the same, so rather than a variance not depending of the mean, one would expect the coefficient of variation to be constant. Furthermore, the phenomena to be modeled are rarely additive in the collateral data. A multiplicative model is much more plausible. If other policy characteristics remain the same, moving from downtown to the country would result in a reduction in the average total claims by some fixed percentage of it, not by a fixed amount independent of the original risk. The same holds if the car is replaced by a lighter one.
The UvA-LINKER will give you a range of other options to find the full text of a publication (inc... more The UvA-LINKER will give you a range of other options to find the full text of a publication (including a direct link to the full-text if it is located on another database on the internet). De UvA-LINKER biedt mogelijkheden om een ...
Multiple linear regression is the most widely used statistical technique in practical econometric... more Multiple linear regression is the most widely used statistical technique in practical econometrics. In actuarial statistics, situations occur that do not fit comfortably in that setting. Regression assumes normally distributed disturbances with a constant variance around a mean that is linear in the collateral data. In many actuarial applications, a symmetric normally distributed random variable with a variance that is the same whatever the mean does not adequately describe the situation. For counts, a Poisson distribution is generally a good model, if the assumptions of a Poisson process described in Chapter 4 are valid. For these random variables, the mean and variance are the same, but the datasets encountered in practice generally exhibit a variance greater than the mean. A distribution to describe the claim size should have a thick right-hand tail. The distribution of claims expressed as a multiple of their mean would always be much the same, so rather than a variance not depen...
In the recent actuarial literature, several proofs have been given for the fact that if a random ... more In the recent actuarial literature, several proofs have been given for the fact that if a random vector X(1), X(2), …, X(n) with given marginals has a comonotonic joint distribution, the sum X(1) + X(2) + … + X(n) is the largest possible in convex order. In this note we give a lucid proof of this fact, based on a geometric interpretation of the support of the comonotonic distribution.
We present a data structure, based upon a hierarchically decomposed tree, which enables us to man... more We present a data structure, based upon a hierarchically decomposed tree, which enables us to manipulate on-line a priority queue whose priorities are selected from the interval 1,..., n with a worst case processing time of (9 (log log n) per instruction. The structure can be used to obtain a mergeable heap whose time requirements are about as good. Full details are explained based upon an implementation of the structure in a PASCAL program contained in the paper. * Work supported by grant CR 62-50. Netherlands Organization for the Advancement of Pure Research (Z.W.O.). 100 P. VAN EMDE BOAS, R. KAAS AND E. ZIJLSTRA
In this paper we examine and summarize properties of several well-known risk measures that can be... more In this paper we examine and summarize properties of several well-known risk measures that can be used in the framework of setting solvency capital requirements for a risky business. Special attention is given to the class of (concave) distortion risk measures. We investigate the relationship between these risk measures and theories of choice under risk. Furthermore we consider the problem of how to evaluate risk measures for sums of non-independent random variables. Approximations for such sums, based on the concept of comonotonicity, are proposed. Several examples are provided to illustrate properties or to prove that certain properties do not hold. Although the paper contains several new results, it is written as an overview and pedagogical introduction to the subject of risk measurement. The paper is an extended version of Dhaene et al. .
Risk measures have been studied for several decades in the actuarial literature, where they appea... more Risk measures have been studied for several decades in the actuarial literature, where they appeared under the guise of premium calculation principles. Risk measures and properties that risk measures should satisfy have recently received considerable attention in the financial mathematics literature. Mathematically, a risk measure is a mapping from a class of random variables defined on some measurable space to the (extended) real line. Economically, a risk measure should capture the preferences of the decision-maker.
A trend in actuarial finance is to combine technical risk with interest risk. If Y t , t ¼ 1, 2, ... more A trend in actuarial finance is to combine technical risk with interest risk. If Y t , t ¼ 1, 2, . . . denotes the time-value of money (discount factors at time t) and X t the stochastic payments to be made at time t, the random variable of interest is often the scalar product of these two random vectors V ¼ RX t Y t . The vectors * X and *
In the present paper we consider several measures Ior the risk that is present in ai1 iiisurance ... more In the present paper we consider several measures Ior the risk that is present in ai1 iiisurance environmeiit. We looi< for clesirable properties for two types of risk ineasures, the olies reflectiiig both negative and positive results, and the ineasures for insolvency risks dealing with aspects of niin, as wel1 as their relatioii to the allocation of ecoiioiiiic capita1 to different brisiness liiies or to the differeiit subcompanies constituting a financial conglomerate. The main probleili for both types of measureii~ents is that the dependence structure that exists betweeii tlie different units involved is uilknown.
In the recent actuarial literature, several proofs have been given for the fact that if a random ... more In the recent actuarial literature, several proofs have been given for the fact that if a random vector (X 1 , X 2 , . . . , X n ) with given marginals has a comonotonic joint distribution, the sum X 1 + X 2 + · · · + X n is the largest possible in convex order. In this note we give a lucid proof of this fact, based on a geometric interpretation of the support of the comonotonic distribution.
A subject often recurring in financial papers, is the pricing of stocks and securities when the r... more A subject often recurring in financial papers, is the pricing of stocks and securities when the rate of return is stochastic. In most cases, the stocks considered are assumed not to pay out any dividend. In the present contribution we want to show how it is possible to obtain upper and lower bounds for the (distribution of the) accumulated value of a dividend paying security at a future time t, when the logarithm of the stock price is modelled by means of a Wiener process.
Comparing risks is the very essence of the actuarial profession. This chapter offers mathematical... more Comparing risks is the very essence of the actuarial profession. This chapter offers mathematical concepts and tools to do this, and derives some important results of non-life actuarial science. There are two reasons why a risk, representing a non-negative random financial loss, would be universally preferred to another. One is that the other risk is larger, see Section 7.2, the second is that it is thicker-tailed (riskier), see Section 7.3. Thicker-tailed means that the probability of extreme values is larger, making a risk with equal mean less attractive because it is more spread and therefore less predictable. We show that having thicker tails means having larger stop-loss premiums.We also show that preferring the risk with uniformly lower stop-loss premiums describes the common preferences between risks of all risk averse decision makers. From the fact that a risk is smaller or less risky than another, one may deduce that it is also preferable in the mean-variance order that is ...
Analysis and Applications in the Social Sciences, 2004
ABSTRACT Multiple linear regression is the most widely used statistical technique in practical ec... more ABSTRACT Multiple linear regression is the most widely used statistical technique in practical econometrics. In actuarial statistics, situations occur that do not fit comfortably in that setting. Regression assumes normally distributed disturbances with a constant variance around a mean that is linear in the collateral data. In many actuarial applications, a symmetric normally distributed random variable with a variance that is the same whatever the mean does not adequately describe the situation. For counts, a Poisson distribution is generally a good model, if the assumptions of a Poisson process described in Chapter 4 are valid. For these random variables, the mean and variance are the same, but the datasets encountered in practice generally exhibit a variance greater than the mean. A distribution to describe the claim size should have a thick right-hand tail. The distribution of claims expressed as a multiple of their mean would always be much the same, so rather than a variance not depending of the mean, one would expect the coefficient of variation to be constant. Furthermore, the phenomena to be modeled are rarely additive in the collateral data. A multiplicative model is much more plausible. If other policy characteristics remain the same, moving from downtown to the country would result in a reduction in the average total claims by some fixed percentage of it, not by a fixed amount independent of the original risk. The same holds if the car is replaced by a lighter one.
The UvA-LINKER will give you a range of other options to find the full text of a publication (inc... more The UvA-LINKER will give you a range of other options to find the full text of a publication (including a direct link to the full-text if it is located on another database on the internet). De UvA-LINKER biedt mogelijkheden om een ...
Multiple linear regression is the most widely used statistical technique in practical econometric... more Multiple linear regression is the most widely used statistical technique in practical econometrics. In actuarial statistics, situations occur that do not fit comfortably in that setting. Regression assumes normally distributed disturbances with a constant variance around a mean that is linear in the collateral data. In many actuarial applications, a symmetric normally distributed random variable with a variance that is the same whatever the mean does not adequately describe the situation. For counts, a Poisson distribution is generally a good model, if the assumptions of a Poisson process described in Chapter 4 are valid. For these random variables, the mean and variance are the same, but the datasets encountered in practice generally exhibit a variance greater than the mean. A distribution to describe the claim size should have a thick right-hand tail. The distribution of claims expressed as a multiple of their mean would always be much the same, so rather than a variance not depen...
In the recent actuarial literature, several proofs have been given for the fact that if a random ... more In the recent actuarial literature, several proofs have been given for the fact that if a random vector X(1), X(2), …, X(n) with given marginals has a comonotonic joint distribution, the sum X(1) + X(2) + … + X(n) is the largest possible in convex order. In this note we give a lucid proof of this fact, based on a geometric interpretation of the support of the comonotonic distribution.
We present a data structure, based upon a hierarchically decomposed tree, which enables us to man... more We present a data structure, based upon a hierarchically decomposed tree, which enables us to manipulate on-line a priority queue whose priorities are selected from the interval 1,..., n with a worst case processing time of (9 (log log n) per instruction. The structure can be used to obtain a mergeable heap whose time requirements are about as good. Full details are explained based upon an implementation of the structure in a PASCAL program contained in the paper. * Work supported by grant CR 62-50. Netherlands Organization for the Advancement of Pure Research (Z.W.O.). 100 P. VAN EMDE BOAS, R. KAAS AND E. ZIJLSTRA
In this paper we examine and summarize properties of several well-known risk measures that can be... more In this paper we examine and summarize properties of several well-known risk measures that can be used in the framework of setting solvency capital requirements for a risky business. Special attention is given to the class of (concave) distortion risk measures. We investigate the relationship between these risk measures and theories of choice under risk. Furthermore we consider the problem of how to evaluate risk measures for sums of non-independent random variables. Approximations for such sums, based on the concept of comonotonicity, are proposed. Several examples are provided to illustrate properties or to prove that certain properties do not hold. Although the paper contains several new results, it is written as an overview and pedagogical introduction to the subject of risk measurement. The paper is an extended version of Dhaene et al. .
Risk measures have been studied for several decades in the actuarial literature, where they appea... more Risk measures have been studied for several decades in the actuarial literature, where they appeared under the guise of premium calculation principles. Risk measures and properties that risk measures should satisfy have recently received considerable attention in the financial mathematics literature. Mathematically, a risk measure is a mapping from a class of random variables defined on some measurable space to the (extended) real line. Economically, a risk measure should capture the preferences of the decision-maker.
A trend in actuarial finance is to combine technical risk with interest risk. If Y t , t ¼ 1, 2, ... more A trend in actuarial finance is to combine technical risk with interest risk. If Y t , t ¼ 1, 2, . . . denotes the time-value of money (discount factors at time t) and X t the stochastic payments to be made at time t, the random variable of interest is often the scalar product of these two random vectors V ¼ RX t Y t . The vectors * X and *
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